Theorem lcfls1lem 35148

Description: Property of a functional with a closed kernel. (Contributed by NM, 27-Jan-2015.)

Hypothesis

Ref	Expression
lcfls1.c	|- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }

Assertion

Ref	Expression
lcfls1lem	|- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )

Distinct variable groups:   f, F   f, G   f, L   ._|_ , f   Q, f

Allowed substitution hint:   C( f)

Proof of Theorem lcfls1lem

Step	Hyp	Ref	Expression
1		fveq2 5892	. . . . . . 7 |- ( f = G -> ( L ` f ) = ( L ` G ) )
2	1	fveq2d 5896	. . . . . 6 |- ( f = G -> ( ._|_ ` ( L ` f ) ) = ( ._|_ ` ( L ` G ) ) )
3	2	fveq2d 5896	. . . . 5 |- ( f = G -> ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) )
4	3, 1	eqeq12d 2477	. . . 4 |- ( f = G -> ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) <-> ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) ) )
5	2	sseq1d 3471	. . . 4 |- ( f = G -> ( ( ._|_ ` ( L ` f ) ) C_ Q <-> ( ._|_ ` ( L ` G ) ) C_ Q ) )
6	4, 5	anbi12d 722	. . 3 |- ( f = G -> ( ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) <-> ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
7		lcfls1.c	. . 3 |- C = { f e. F | ( ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) /\ ( ._|_ ` ( L ` f ) ) C_ Q ) }
8	6, 7	elrab2 3210	. 2 |- ( G e. C <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
9		3anass 995	. 2 |- ( ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) <-> ( G e. F /\ ( ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) ) )
10	8, 9	bitr4i 260	1 |- ( G e. C <-> ( G e. F /\ ( ._|_ ` ( ._|_ ` ( L ` G ) ) ) = ( L ` G ) /\ ( ._|_ ` ( L ` G ) ) C_ Q ) )

Syntax hints:   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   e. wcel 1898   {crab 2753   C_ wss 3416   ` cfv 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-iota 5569  df-fv 5613

This theorem is referenced by:  lcfls1N  35149  lcfls1c  35150  lclkrslem1  35151  lclkrslem2  35152  lclkrs  35153